Calculus for Business & Economics Notes

Math 140 - Spring 2025

Week 1 Notes

Mon, Jan 13

Today we introduced the course syllabus. Then we looked at how to solve the following equations algebraically.

1. Solve $5-2x = -5$. (video)

2. Solve $\sqrt{3x + 1} = 4$. (video)

3. Solve $\dfrac{x}{x-1} = 3$.

After we solved these equations, we also talked about the geometric interpretation of the solutions as places on a graph where a function has a certain y-value.

4. Graph the line $y = 5 - 2x$ and check where it crosses the line $y = -5$.

5. Graph the function $y = \sqrt{3x + 1}$.

Wed, Jan 15

To make it easier to graph functions, it helps to know some basic graphs. Here are six you should memorize.

We used these examples to help graph the following in class:

1. $y = 5|x|$

2. $y = x^2 + 3$ (video)

3. $y = \dfrac{1}{x-2}$ (video)

4. $y = \sqrt{4 - x}$

After those graphs, we talked about function notation. Both $y = x^2$ and $f(x) = x^2$ to mean the same thing. But the notation $f(x) = x^2$ emphasizes that $f$ is a function of the $x$ variable. Be careful not to confuse function notation $f(5)$ with multiplication $(2)(5) = 10$. Even though the notation looks the same, they are not the same!

We did these examples.

5. If $f(x) = x^2$ and $g(x) = x+5$, find $f(g(4))$ and $g(g(4))$.

6. Find $3 f(2) + 4 g(-1)$.

7. The quantity of gasoline $Q$ sold by a gas station is a function of the price $p$ that the owner sets. Here is a graph of the function $Q = Q(p)$.

   

   1. Use the graph above to find $Q(3)$.
   2. Solve $Q(p) = 3000$ for $p$.
      
   3. If $Q(5) = 600$, explain in English what that means.

8. The function $f(x) = \frac{1}{2}(x + \frac{2}{x})$ can be used to approximate $\sqrt{2}$. Calculate $f(2)$ and $f(f(2))$.

Fri, Jan 17

We started talking about different ways you can combine functions. We did the following exercises.

1. Use the graph below to compute $g(f(-5))$. (video)

2. (Exercise 1.2# 29) The function $D(p)$ gives the number of items that will be demanded when the price is $p$. The production cost, $C(x)$ is the cost of producing $x$ items. To determine the cost of production when the price is $6, you would do which of the following?

   1. Evaluate $D(C(6))$
   2. Evaluate $C(D(6))$
   3. Solve $D(C(x)) = 6$
   4. Solve $C(D(p)) = 6$

3. Continuing the previous problem, profit is revenue minus cost, and revenue is price times quantity sold. Using the functions $C$ and $D$, write down formulas for revenue and for profit.

After we talked about function composition, we switched to a quick review of linear functions. You need to know these formulas for linear functions:

• Slope-Intercept Form $y = mx + b$
• Point-Slope Form $y-y_0 = m(x - x_0)$

You also need to understand slope very well:

• Slope $m = \dfrac{\text{rise}}{\text{run}} = \dfrac{\Delta y}{\Delta x} = \dfrac{ y_2 - y_1}{x_2 - x_1}$

4. A line passes through $(-1, 6)$ and $(5, -4)$. Find an equation for the line. (video)

Week 2 Notes

Wed, Jan 22

Today we talked about linear equation word problems.

1. Someone is hiking up a mountain. They start at an elevation of 1200 meters and climb at a constant rate. After 4 hours, they are at an elevation of 1700 meters. Find a formula for their elevation $y$ as a function of the time $x$ in hours since they started hiking. (video)

2. Find a formula to convert Celsius to Fahrenheit.

3. Find a formula to convert Fahrenheit to Celsius.

4. Find the slope and $y$ intercept of the equation $5x-2y = 8$. (video)

5. Joe imports coffee. He can import arabica beans for $6 per pound and he can import robusta beans for $5 per pound. Suppose he spends $1500 to import $x$ pounds of arabica beans and $y$ pounds of robusta beans.

   1. Find an equation that relates $x$ and $y$.
   2. What is the $x$-intercept for this equation, and what does it mean?
   3. What is the $y$-intercept for this equation, and what does it mean?
   4. What is the slope of this equation, and what does it mean?

6. The Virginia state income tax for people making over $17,000 is $720 plus 5.75% of all income over $17,000. Find a formula for the tax owed $y$ as a function of income $x$.

Fri, Jan 24

1. Solve the system of equations below. (video)

$$y = 2x$$ $$3x + y = 30$$

2. Solve the system of equations below. (video)

$$3x + 6y = 12$$ $$5x - 8y = 2$$

3. Suppose that the quantity of beef supplied ($Q_S$) and demanded ($Q_D$) are given by the following functions of price. Find the equilibrium price and quantity. (video)

$$Q_S = -40 + 30P$$ $$Q_D = 180 - 25P$$

4. Leo has 4 more keys than Haley on his keychain. Together they have 18 keys total. How many keys does Haley have? (video)

5. Solve the system of equations

$$y = \dfrac{6}{x}$$ $$y = 2 - \dfrac{5}{x}$$

Week 3 Notes

Mon, Jan 27

Today we started talking about exponent rules. There are a bunch of rules, but they all boil down to three simple ideas:

1. Simplify $\left( \dfrac{x^7 y^3}{x^2 y} \right)^4$. (video)

2. Simplify $(a^3 b)^4 (a b^6)^2$. (video)

3. Simplify $(b^{-3} c^{-7})^{-2} (b^3 c^{-2})^{-3}$. (video)

4. Simplify $16^{1/4} + 16^{-1/4}$.

5. Simplify $\sqrt{x^2 + x^2 + x^2 + x^2}$. What do you think the most common mistake is here?

6. Simplify $4^{5/2}$.

7. Simplify $\sqrt[3]{64 z^6}$. (video)

Wed, Feb 29

Today we looked at using the exponent rules to help solve equations.

1. Solve $(x^{-1} + 2^{-1})^{-1} = \frac{1}{3}$.

2. Solve $\dfrac{2^{57}}{2^x} = 8$.

3. Solve $3^{x+2} = 9^{2x - 3}$. (video)

4. Solve $x^{-5/3} = \frac{1}{32}$. (video)

5. Solve $2^{\sqrt{x}} = 8^x$. (video)

This last problem opens a can of worms, since it turns into a 2nd degree polynomial. A polynomial is any sum of terms that combine a whole number power of $x$ multiplied by a number called a coefficient.

Solving equations involving polynomials requires a non-intuitive technique:

Here is another polynomial equation:

6. Solve $10x^2 - 6x = 0$. (video)

Fri, Jan 31

Here’s an example we didn’t get to last time.

1. Solve $\sqrt{3x + 36} + 6 = x$. (video)

After some algebra, this turns into the equation $x^2 - 15x = 0$. We can factor that by dividing out the common factor of $x$. There is a second more complex kind of factoring that involves un-FOIL-ing a polynomial. Recall that FOIL stands for First-Outside-Inside-Last, which is a mnemonic to remember how to multiply out expressions like: $$(x+a)(x+b) = x^2 + bx + ax + ab = x^2 + (a+b)x + ab.$$ When you factor a polynomial with leading term $x^2$, you need to find factors of the constant term that add up to the middle coefficient. Here are some examples we did in class.

2. Factor $x^2 + 15x + 50$. (video)

3. Factor $x^2 - 11x + 24$. (video)

4. Solve $x^2 = 3x + 10$. (video)

We didn’t get to these last two examples in class, but they are good extra practice:

5. Where does the line $y = x+1$ intersect the parabola $y = x^2-1$? (video)

6. Solve $(x+2)(x-5) = 8$

Week 4 Notes

Mon, Feb 3

We did this in-class workshop:

• Workshop: Parabolas

Wed, Feb 5

We looked at some more examples of factoring to solve polynomial equations.

1. $-0.1x^2 + 0.7x + 6 = 0$. (This was from the workshop last time.)

2. $4x^6 + 4 x^5 - 24 x^4 = 0$. (video)

A rational function is a function that can be expressed using polynomials and reciprocals of polynomials. You can use the same techniques we’ve already used to solve equations involving rational expressions.

3. $x + 2 = \dfrac{15}{x}$.

After that, we talked about how to solve inequalities. We started with simple examples like:

4. Solve $x + 4 > 3$ and $-2x \ge 4$.

Then we moved on to harder examples.

4. Solve $x^3 - 3x^2 < 4x$. (video)

5. Solve $\dfrac{2}{x-3} \ge 1$. (video)

6. Solve $x+3 > \dfrac{4}{x}$.

Fri, Feb 7

Today we talked about exponential growth. We started with exponential functions.
$$f(x) = C_0 b^x$$ where $C_0$ is the initial value and $b > 0$ is a constant called the base. The most common bases are base-10, base-2, and the natural exponential base which is $e = 2.71828\ldots$ which is also known as Euler’s number.

1. Graph the function $y = 2^x$.

Exponential growth/decay is what you get when you multiply by the same growth factor every step. This is different than linear growth where you add the same constant rate of change every step.

2. For each sequence below, determine if it a linear or exponential pattern. If it is linear, what is the rate of change? If it is exponential, what is the growth factor?

   1. $3, 5, 7, 9, 11, \ldots$
   2. $400, 200, 100, 50, 25, \ldots$
   3. $18, 12, 8, \tfrac{16}{3}, \tfrac{32}{9}, \ldots$

3. According to Newton’s law of cooling, the temperature difference between a small object and its surroundings changes exponentially. Suppose a cup of coffee is initially $60^\circ$C above room temperature. If its temperature has decreased to only $50^\circ$C above room temperature ten minutes later, then what will the temperature be after 20 minutes? What about after 30 minutes? What is the growth factor?

4. Suppose that the median house price in a city has been growing exponentially for some time. If the median price was $\$200{,}000$ in the year 2000 and $\$260{,}000$ in 2010, then what were prices in 2020? What are prices now? Find an equation for the price as a function of time.

Week 5 Notes

Mon, Feb 10

One of the most common ways to talk about growth is with percentages. This type of growth is called relative growth because it is stated relative to the size of the population that is growing with a percentage. For example, 5% growth of the United States population would be a lot, because the US population is large. The best way to understand relative growth is to convert percentage increases and decreases into growth factors.

$$\mathbf{growth~factor} = 100\% + \text{percent change}.$$

A growth factor is also the ratio of the new amount divided by the old amount.

$$\mathbf{growth~factor} = \dfrac{\text{ new amount }}{\text{ old amount }}.$$

1. The US population was 310 million in 2010. It was 330 million in 2020. What is the growth factor and the percent change in the US population from 2010 to 2020?

2. What would the growth factor and percent change be if the population were to decline from 330 million in 2020 to 320 million in 2030?

The tricky thing about percent growth is that it is not additive. If something grows by 30% for two years in a row, that is not 60% total growth. You have to multiply the growth factors.

3. If something grows by 30% one year and then 30% again the next year, how much did it grow total as a percentage?

4. If I have $100 invested and its value grows by 25% in the first year and shrinks 25% in the second year, then how much money is left after two years? (video)

Since you multiply growth factors instead of adding percent changes, you also have to take roots to break growth factors into equal amounts.

5. What was the (average) annual percent population change in the United States between 2010 and 2020?

6. If that rate of population growth continues, what is the equation for the population of the United States as a function of time?

7. If a savings account earns 2% interest every year, and you put $100 in the bank and leave it there for 10 years, how much money will you have? What about after $x$ years?

8. Which is better, to invest in a mutual fund that grows by 10% every year, or a fund that grows by 0.8% every month?

Here is one more example that we didn’t have time for in class:

9. Last time we talked about a cup of coffee that cools exponentially. If it cools from $60^\circ$C above room temperature to $50^\circ$C in 10 minutes, then what is the percent change in temperature every minute?

Wed, Feb 12

Today we went over the midterm 1 review problems and a few problems from homework 5.

Week 6 Notes

Mon, Feb 17

Today we introduced logarithms. We started by talking about logarithmic scales, which are number lines where the numbers are spaced by multiplication instead of addition.

Logarithmic scales help understand the logarithm function.

1. Find $\log_2 16$. (video)

2. Find $\log_3 81$. (video)

3. Find $\log_{100} 1$. (video)

4. Find $\log_2 \left( \frac{1}{64} \right)$. (video)

5. Find $\log_{10} (\sqrt{10}).$

Logarithms are useful because they convert difficult multiplication/division problems into easier addition/subtraction problems. They also convert exponential patterns into linear patterns.

The most important base for exponential and logarithmic functions in calculus is the number $e \approx 2.71828$. This is the natural base for the logarithm and exponential function for reasons that we’ll see later when we talk about derivatives. We write $\ln(x)$ to denote the base-e logarithm.

6. What are $\ln(e^x)$ and $e^{\ln x}$?

One of the most important applications of logarithms is that they let you solve equations with variables in the exponent.

7. Solve $e^x = 2$.

8. Solve $3^x = 2^{x+1}$ (video)

Here is one more good exercise using logarithms that we didn’t have time for:

9. A lumber company has 1,200,000 trees. They plan to harvest 7% of the remaining trees each year. How long until they have harvested half of the trees? (video)

Wed, Feb 19

Class is canceled today because of the snow, but I recommend starting homework 6 if you haven’t already. Here are some video links for problems similar to the ones on the homework.

1. Which of the following equations would you need a logarithm to solve? (<https://www.youtube.com/shorts/2MT89tcjnQo))

   1. $3^x = 9$
   2. $3^x = 27$
   3. $3^x = 10$

2. Solve $10^{2t-3} = 7$. (video)

3. Solve $3^x = 7^{x+2}$. (video)

4. How long would it take an investment of $4,000 to grow to $20,000 if it grows by 7% each year? (video)

5. A lumber company has 1,200,000 trees. They plan to harvest 7% of the remaining trees each year. How long until they have harvested half of the trees? (video)

Fri, Feb 21

We started by going over problems #5 and #7 from Homework 6. We also talked about the shape of the graph of a logarithm function. You should memorize the graph of the natural exponential and logarithm functions.

After that quick review, we introduced derivatives.

Calculus was discovered independently by both Newton and Leibniz, and they invented different notations for the derivative. Newton used a mark like $y'$ or $f'(x)$ to represent the derivative. Leibniz used the fraction $dy/dx$. Newton’s notation emphasizes that the derivative is a function that depends on which $x$ we input. Leibniz’s notation emphasizes that the derivative is the slope of a tangent line, so it is equal to a rise over a run. The symbols $dy$ and $dx$ are called differentials, and can represent any rise and run of a tangent line, the same way that $\Delta y$ and $\Delta x$ represent rise and run for other lines. (video)

1. If you drop a rock from a height of 400 feet in the air, its height after $t$ seconds will be $$h(t) = -16 t^2 + 400$$ The derivative of height is the velocity: $$h'(t) = -32 t.$$

   1. What is the average velocity of the rock as it falls?

   2. Zoom in on the graph at $t = 5$ seconds. Use the graph to estimate the instantenous velocity when the rock hits the ground.

   3. Guess which point on the graph has a tangent line with slope equal to the average velocity over the interval $[0,5]$.

2. Graph the velocity function $v(t) = -32t$. Why are the average rate of change and instantaneous rates of change the same for the velocity function? What is the derivative of the velocity, and what does it represent?

Week 7 Notes

Mon, Feb 24

Today we introduced some rules for calculating the derivative. For any function $f(x)$, the notation $$\dfrac{d}{dx} f(x)$$ means “take the derivative of the function”. We write either $f'(x)$, $y'$, $\frac{dy}{dx}$, or sometimes $\frac{df}{dx}$ to represent the result.

1. Use the power rule to find each of the following derivatives.

   1. $\frac{d}{dx} x^2$
   2. $\frac{d}{dx} x^3$
   3. $\frac{d}{dx} x^4$
   4. $\frac{d}{dx} x^{-1}$
   5. $\frac{d}{dx} \sqrt{x}$

2. What is the slope of the tangent line to $y = \sqrt{x}$ at the point $(4,2)$?

3. Find the derivative of $y = 6x^2$.

4. Find the derivative of $f(x) = 4x - x^3$.

5. In exercise 4 from the Parabolas workshop, we looked at an example where a gas station’s revenue $R$ is a function of the price they charge $p$ per gallon. The formula for the revenue was $$R(p) = 36000p - 9000p^2$$

   1. Find the formula for the derivative $\frac{dR}{dp}$.
   2. Find the value of $\frac{dR}{dp}$ when $p = \$3$ and explain what it means.

6. Find the derivative of $y = (x-1)(x-4)$.

Wed, Feb 26

Class was canceled today.

Fri, Feb 28

We applied the rules from last time to find the derivatives of the following examples.

1. Let $f(x) = x^3 - 6x^2 + x - 5$. Find the slope of the tangent line when $x = 1$. (video)

2. Differentiate $y = 7x ( 2x - x^3)$. (video)

3. Find $\dfrac{d}{dx} \dfrac{4x^5 - 5x^4 + 2x^3}{x^2}$. (video)

4. Find $\dfrac{d}{dx} \sqrt[3]{x^2}$. (video)

5. Find the derivative of $y = \dfrac{\sqrt{x}}{x^2}$.

In addition to these exercises, we also talked about derivative notation.

The derivative can be a formula or a number. So it might make sense to say that $\dfrac{dy}{dx} = 3x^2$, but it would never make sense to say that $\dfrac{d}{dx} = 3x^2$.

Week 8 Notes

Mon, Mar 3

Today we talked about applications of derivatives. One application is to find the points on a graph where the slope is zero:

1. Let $p(x) = \frac{1}{3}x^3 - x^2 - 8x + 24$. Find $p'(x)$ and find the points where $p'(x) = 0$.

2. Let $y = \dfrac{1}{2x^2} - \dfrac{6}{x}$. Find $y'$ and the point where $y' = 0$.

Another important application in economics is the notion of marginal functions including marginal cost and marginal revenue. If a company produces $x$ goods and the cost to produce those goods is a function $C(x)$, then the marginal cost to produce the next one item after already producing $x$ is called the marginal cost. Technically it is equal to $C(x+1) - C(x)$, but it is often easier to just use the derivative $C'(x)$ to estimate the marginal cost instead. Likewise, the marginal revenue is the extra bit of revenue that comes from selling the next one item after selling the first $x$, and it can be approximated by the derivative $R'(x)$ of the revenue function.

3. Suppose the cost of producing $x$ machines is $C(x) = 1100 + 140x - 0.2x^2$. Find the marginal cost of producing one more machine beyond $x = 105$. Show that this is roughly the same as the actual cost of the 106th machine. (video)

Wed, Mar 5

We started with some more marginal analysis examples.

1. Suppose a company has total revenue $R(x) = 30x - 0.025x^2$ and cost $C(x) = 2x + 5$. Find the marginal revenue, marginal cost, and marginal profit. (video)

2. Suppose that $C(x) = \frac{100}{x} + 0.05x^2$. Find the marginal cost. When is the marginal cost zero?

Although derivatives work term-by-term, they don’t play nice with factors. For example, you can’t just take the derivatives the two factors in the expression $x^2 \cdot x^3$. To work with factors, you need to use the product rules.

We did the following examples:

3. $\dfrac{d}{dx} x^2 ( x^3 + 4)$ (video)

4. $\dfrac{d}{dx} (9x^2 - 6x + 2) e^x$ (video)

5. $\dfrac{d}{dx} x^{-5} e^x$ (video)

6. $\dfrac{d}{dx} (x-1) \ln x$ (video)

In many cases, we can also use the rules for exponents and logarithms to simplify functions before we calculate the derivative.

7. $\dfrac{d}{dx} \ln(5x^2)$.

8. $\dfrac{d}{dx} \ln(3^{5x})$ (video)

Fri, Mar 7

We started with an abstract product rule example.

1. Suppose that $f(x) = g(x) h(x)$ where the values of $g$ and $h$ at $x=5$ are given by the table below. Find the value of $f'(5)$. (video)

Then we introduced the quotient rule.

2. Find the derivative of $y = \dfrac{4x^2 - 3}{2x + 1}$. (video)

3. Find $\dfrac{d}{dx} \dfrac{\ln x}{ x }$.

4. Use the quotient rule to find $\dfrac{d}{dx} e^{-x}$. (video)

5. Suppose that $f(x) = \dfrac{g(x)}{h(x)}$ where the values of $g$ and $h$ at $x=5$ are given by the table below. Find the value of $f'(5)$.

Week 9 Notes

Mon, Mar 17

We started with some practice examples to review the product and quotient rules.

1. $f(x) = \dfrac{5x+2}{3x-4}$. Find $f'(x)$. (video)

2. $\dfrac{d}{dx} 3x^4 (2x^2 - 1)$. (video)

3. $\dfrac{d}{dx} \dfrac{1}{x^2 + 4x + 4}$

After those examples, we introduced the chain rule, which is the last and also one of the most important rules for finding the derivative.

The chain rule takes practice to get used to, but here is an intuitive example to start to get the hang of it.

1. If you drive along Interstate-70 across Kansas from milepost 0 in the west to milepost 400 near where the highway crosses into Missouri, then your elevation decreases from 1270 meters to only 270 meters. So the average change in elevation is $-2.5$ meters per mile as you drive east. How fast is your elevation changing if you are driving at 60 miles per hour?

In this example, $x$ is miles driven, $y$ is elevation, and $t$ is time. If you are driving 60 miles per hour, then $x(t) = 60t$. And the elevation is $y(x) = 1270 - 2.5 x$. To find $x'(t)$, you need to multiply $y'(x(t)) \cdot x'(t)$.

2. $y = (4x+3)^3$, find $y'$. (video)

3. $\dfrac{d}{dx} \ln( x^2 + 1 ).$ (video)

4. Find $\dfrac{d}{dx} (x^2 + 3x)^{100}$.

Wed, March 19

Today we did more examples of the chain rule.

1. If $y = e^{-x^2}$, find $\dfrac{dy}{dx}$. (video)

2. $\dfrac{d}{dx} \ln( x + 2 )$.

You can also use the chain rule to avoid using the quotient rule. Here is an example we did last time, re-written as a product rule:

3. Find the derivative of $\dfrac{5x + 2}{3x - 4} = (5x+2)(3x - 4)^{-1}.$

4. A rock is thrown into the center of a still pond, causing ripples to spread out in a circle. The ripples move outwards at 4 feet per second.

   1. Find a formula for the area of the disturbed water as a function of radius.
   2. Find a formula for the area of the disturbed water as a function of time.
   3. Calculate $\dfrac{dA}{dr}$
   4. Calculate $\dfrac{dA}{dt}$

5. Suppose that the crime rate $C(p)$ in a city is a function of the population $p$. The population $p=p(t)$ is a function of time $t$ in years. Suppose that the city’s population is currently 300,000 at $t=0$. If $p'(0)=5{,}000$ people per year and $C'(300{,}000)=0.4$ crimes per person, then estimate the rate of change in crime this year.

Here are some examples that combine the chain rule with other rules:

6. Find the derivative of $\left(\dfrac{x-1}{2x+1} \right)^{7}$. (video)

7. $\dfrac{d}{dx} x^2 \sqrt{4 - 9x}$. (video)

Fri, Mar 21

Why is the chain rule called the chain rule? It’s because you can apply the chain rule to a sequence of nested functions, no matter how long the chain of functions is. For example: if $z = z(y)$, $y = y(x)$, and $x = x(t)$, then this chain of functions has derivative: $$\frac{dz}{dt} = \frac{dz}{dy} \frac{dy}{dx} \frac{dx}{dt}.$$

Here is an example of a problem where you need to use the chain rule twice:

1. $y = e^{(3x+2x^2)^5}$, find $\dfrac{dy}{dx}$. (video)

After that, we talked about higher derivatives. The second derivative of a function $y = f(x)$ is $$y'' = \frac{d}{dx} \left( \frac{d}{dx} y \right) = \dfrac{d^2}{dx^2} y = \dfrac{d^2 y}{dx^2} = f''(x).$$

The first derivative is the slope of the tangent line.

2. Find the second derivatives of $\ln x$ and $e^x$. Does the value of the second derivative match the concavity of the graphs for these functions?

3. Find the second derivative $y''$ when $y = \dfrac{6}{x^2}$. (video)

Week 10 Notes

Mon, Mar 24

In physics, if $s(t)$ is the position of an object as a function of time, then the first derivative is velocity and the second derivative is acceleration.

1. A rock thrown straight up has height $h(t) = 4 + 30t - 16t^2$ in feet after $t$ seconds. Find $h'(t)$ and $h''(t)$ (that is, find the velocity and acceleration).

2. The normal distribution (bell curve) in statistics has equation $y = e^{-x^2/2}$. Find the first and second derivatives, and use the second derivative to determine when the graph is concave up and concave down.

3. Find the intervals where the function $f(x) = \dfrac{x^4}{8} - \dfrac{x^3}{2}$ is concave up. (similar example)

4. Find the inflection points of $g(x) = \tfrac{1}{4} x^4 - 4x^3 + 24x^2$. (video)

Here are some additional problems that we didn’t have time for in class.

5. Find the intervals where the function $y = \dfrac{x^{4}}{12} - \dfrac{x^{3}}{3} - \dfrac{3 x^{2}}{2}$ is concave up.

6. Find the inflection points of $y = \ln(1+x^2)$. (video)

Wed, Mar 26

Today we went over the midterm 2 review problems in class.

Week 11 Notes

Mon, Mar 31

We started by reviewing the problem on the midterm exam where the temperature of a can of soda in Celsius after $t$ hours in a refrigerator is $$T(t) = 4 + 20 e^{-3t}.$$ We talked about how to find the derivative and about what the derivative means. In particular we talked about units and how to find the units of derivatives.

After that, we talked about finding the intervals where a function is increasing and where it is decreasing. This works exactly like finding the intervals of concavity, except you use the first derivative, not the second. We also talked about local maximums and local minimums. A local max occurs when a continuous function is increasing before and decreasing after a point. A local minimum occurs when the function is decreasing before and increasing after a point.

1. Find the intervals of increase and decrease for $f(x) = x^2 - 3x + 1$. (video)

2. Find the intervals of increase and decrease for $y = x^3 - 9x^2 + 24x$. (video)

3. Find the intervals of increase and decrease for $y = \dfrac{4}{x} + x$.

Wed, Apr 2

We continued doing optimization problems today. We did these examples.

1. Find the local max and mins for $y = \dfrac{x^{5}}{5} - \dfrac{x^{4}}{4} - 2 x^{3}$.

2. Find the intervals of increase and decrease for $f(x) = \dfrac{x^2}{x^2 - 1}$. (video)

We also talked about finding the absolute maximum and minimum y-values on an interval. The key idea is that you need to check the y-values at both the critical points inside the interval and also the endpoints of the interval. We did these two examples:

3. Find the absolute max and min of $y = -x^2 + 4x - 3$ on $[0,4]$.

4. Find the absolute max and min of $y = xe^{-x}$ on $[0,2]$. (video with similar example)

Fri, Apr 4

Today we went over homework 10. We also looked at this example:

1. Find absolute max and min of $f(x) = 8 \ln x - x^2$ on the interval $[1,4]$. (video)

Then we looked at examples of optimization word problems.

2. A farmer wants to fence off a rectangular plot of land along the side of a long straight river. He has a total of 2400 feet of fence. How large of and area can he fence off? (video)

With the last example, we talked about the first and second derivative tests.

Week 12 Notes

Mon, Apr 7

A differential represents a change in a variable along a tangent line, and so it can have a numerical value. The change in $y$ along the tangent line is $dy$ and the change in $x$ is $dx$.

1. Find the differential $dy$ when $y = x^2$. What is $dy$ when $x = 2$ and $dx = 0.1$?

2. Find the differential $dy$ when $y = e^{x/10}$. What is the value of $dy$ when $x = 10$ and $dx = 0.05$? (video)

3. The radius of the Earth is about 4000 miles. According to the snowball Earth hypothesis, the Earth might once have been covered in a layer of ice that was half a mile thick. Approximately how many cubic miles of ice is that? Hint: Find the differential of $V = \tfrac{4}{3} \pi r^3$ when $r = 4{,}000$ and $dr = 0.5$.

4. Estimate $\sqrt{101}$ using differentials. Hint: How much bigger is $\sqrt{101}$ than $\sqrt{100}$? Calculate the differential of $y = \sqrt{x}$ at $x = 100$ to find out. (video)

Wed, Apr 9

Today we looked at examples of functions of two or more variables and how to graph them. There are two options for graphing function with two variables. You can either graph the level curves on a regular xy-plane or you can graph the surface $z = f(x,y)$ using a 3D grapher (like Desmos 3D).

We discussed the following examples:

1. Volume of a cylinder is $V(r, h) = \pi r^2 h$.

2. Body mass index (BMI) is $$\operatorname{BMI} = 703 \frac{W}{h^2}$$ where $W$ is a person’s weight in lbs and $h$ is their height in inches. A BMI less than 18.5 is underweight, over 25 is overweight, and over 30 is obese. Find $\operatorname{BMI}(180 \text{lbs.}, 70 \text{in.})$.

3. Volume of a box is $V(l,w,h) = lwh$.

4. We made a 3D graph of the function $f(x,y) = 4-x^2 -y^2$ using (Desmos 3D).

5. Make a graph showing three different level curves for $g(x,y) = x^2 + y$. (Desmos)

6. Find $\dfrac{\partial g}{\partial x}$ and $\dfrac{\partial g}{\partial y}$ for $g(x,y) = x^2 + y$.

7. Find $\dfrac{\partial V}{\partial r}$ and $\dfrac{\partial V}{\partial h}$ for $V = \pi r^2 h$.

8. Find $f_x$ and $f_y$ for $f(x,y) = x e^{-2x}$. (video with similar example)

Fri, Apr 11

We spent most of class reviewing homework 11 before the quiz today. But at the end, we looked at this example of partial derivatives.

1. Find $\dfrac{\partial}{\partial x} x^2 y^2 + x + y^4 + 5$. Then find the partial derivative with respect to $y$. (video)

Week 13 Notes

Mon, Apr 14

We started with this warm-up problem.

1. Find the partial derivatives of $z = \sqrt{9 - x^2 - y^2}$ when $x = 2$ and $y = 1$.

2. The body mass index function is $B = \dfrac{m}{h^2}$ when $m$ is a person’s mass in kilograms and $h$ is their height in meters. Find the partial derivatives $\dfrac{\partial B}{\partial m}$ and $\dfrac{\partial B}{\partial h}$ for someone who is 2 meters tall (about 6’6”) and weighs 100 kg (about 220 lbs.). Include the units for both partial derivatives.

3. A factory produces $Q = 60 K^{1/2} L^{1/3}$ units of output where $K$ is the capital invested in the factor (in dollars) and $L$ is the worker-hours of labor. Find $Q_K$ and $Q_L$ when $K = \$9{,}000{,}000$ and $L = 1000$ hours of labor. Which would increase production more, adding one extra hour of labor or $1000 of capital investment?

A special kind of multivariable function is a linear function of two or more variables. A linear function of two variables has the form $$z = ax + by + c$$ where $a$, $b$, and $c$ are constants. The graph of a linear function with two input variables is a flat 2-dimensional plane in 3-dimensional space.

4. What are the partial derivatives of a linear function $z = ax + by + c$?

An important thing to understand about partial derivatives is that they let you find the direction of steepest ascent also known as the gradient of a function. If you start at a point $(x_0, y_0)$, then draw an arrow by moving $\dfrac{\partial f}{\partial x}$ units to the right and $\dfrac{\partial f}{\partial y}$ units upwards, then that arrow always points in the direction of steepest ascent. It is always perpendicular to the level curve at $(x_0, y_0)$.

Week 14 Notes

Mon, Apr 21

Today we talked about how to find the max or min of a function with two variables. Since there are two partial derivatives, you have to set them both equal to zero to find the critical points.

1. $f(x,y) = x^2 + xy + y^2 - 3x$ (video)

2. $z = 6x+8y - x^2 - y^2$.

3. $z = x^3 + y^2 - 12x$. Notice that this one has two different critical points.

Wed, Apr 23

1. A company makes two products. The demand equations for the two products are given below where $x$ and $y$ are the prices the company chooses for product 1 and product 2 respectively. $$Q_1=200-3x-y$$ $$Q_2=150-x-2y$$ Find the price the company should charge for each product in order to maximize total revenue. What is that maximum revenue?

Many multivariable optimization problems have both an objective function that you want to maximize or minimize and a constraint equation. One method for solving these problems is to use Lagrange multipliers.

The intuition behind these equations is that the direction of greatest ascent at the max/min point (blue arrow in the picture below) points in the same (or exact opposite) direction as the direction of greatest ascent for the constraint equation (red arrow). So there is a number $\lambda$ (called the Lagrange multiplier) such that the coordinates of the blue arrow are just $\lambda$ times the coordinates of the red arrow.

We used the method of Lagrange multiplier to solve this problem in class:

2. According to US postal service rules, a cylindrical package may not have a circumference plus height of more than 9 feet. Find the dimensions of a cylindrical package that you could mail with the largest volume.

Here are two more examples that we didn’t do in class, but they both have video explanations.

3. Use the Lagrange multiplier method to maximize $f(x,y) = 3x + 4y$ subject to the constraint $x^2 + y^2 = 100$. (video)

4. Suppose that the level of production $Y = 32 L^{0.6} K^{0.4}$ where $L$ is the amount of labor (in worker-hours) and $K$ is the capital invested (in dollars). Find the maximum level of production given the constraint $4 L + 2 K = 50$. (video)

Fri, Apr 25

Today we went over homework 12 before the quiz.